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Home > Binomials

Learn more about "Binomials"

Binomial

In elementary algebra, a '''binomial''' is a polynomial with two terms—the sum of two monomials—often bound by parenthesis or brackets when operated upon. It is the simplest kind of polynomial other than monomials.

Operations on simple binomials

- The binomial

a^2 - b^2

can be factored as the product of two other binomials:
- :

a^2 - b^2 = (a + b)(a - b).

- This is a special case of the more general formula

a^{n+1} - b^{n+1} = (a - b)\sum_{k=0}^{n} a^{k}\,b^{n-k}

.
- The product of a pair of linear binomials

(ax+b)

and

(cx+d)

is:
- :

(ax+b)(cx+d) = acx^2+axd+bcx+bd

- A binomial raised to the ''n^th'' Exponentiation|power, represented as
- :

(a + b)^n

- can be expanded by means of the binomial theorem or, equivalently, using Pascal's triangle. Taking a simple example, the perfect square binomial

(p+q)^2

can be found by squaring the :first digit, adding twice the product of the first and second digit and finally adding the square of the second digit, to give

p^2+2pq+q^2

.
- A simple but interesting application of the cited binomial formula is the "(m,n)-formula" for generating Pythagorean triples: for ''m < n'', let

a=n^2-m^2,\ b=2mn,\ c=n^2+m^2

, then

a^2+b^2=c^2

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